Real Life Applications: Matthew Aguilar
Notes 4-5: Katie Conner
Example Problems: Cody Windsor
Vocab: Austin Gibbons
editor/ video: Cheyenne Bohlen notes 1-3:Haley Brandt
A matrix is a rectangular arrangement of numbers in rows ad columns. For instance, matrix A below has two rows and three columns. The dimensions of this matrix are 2x3 (read 2 by 3). The numbers in a matrix are its entries. In matrix A, the entry in the second row, third column is 5.
Properties of matrix operations Let A, B, and C be matrices of the same dimensions and let d be scalar.
Associative Property of Addition (A+B)+C=A+(B+C) Commutative Property of Addition A+B=B+A
Multiplication of a sum or difference of a matrix follows the distributive property.
Distributive Property of Addition d(A+B)=dA=dB Distributive Property of Subtraction d(A-B)=dA-dB
Multiplying Matrices The product of two matrices is only defined if the number of columns inA is equal to the number of rows inB. IfAhas a dimension ofmxn andBhad a dimension of nxpthen the product ofABis mxpmatrix.
There is a specific pattern for multiplying matrices.The first number of the first row of theAmatrix is multiplied by the first number of the first column of the Bmatrix. Next multiply the second number of the first row of the A matrix by the second number of the first column of the B matrix. Then add the products ae and bg. This is the first entry in your new matrix, C. ae+bg=i Next multiply the first number of the first row in matrix A by the first number in the second column of matrix B. Then multiply the second number of the first row in matrix A by the second number in the second row of matrix B. Add the products af and bh. This becomes the second number of the first row in your new C matrix. af+bh=j Take the first number of the second row of matrix A and multiply it by the first number in the first column of matrix B Next take the second number of the second row of matrix A and multiply it by the second number of the first column of the B matrix. Next take the product ce and add it to dg. This is the third number in your new matrix. Take the first number of the second row of matrix A and multiply it by the first number of the second column of matrix B. Take the second number of the second row of matrix A and multiply it by the second number of the second column of matrix B. Now add the products cf and dh. This is the last number of your matrix.
Multiplication Properties of matrices
Assosiative Property of matrix multiplication A(BC)=(AB)C Left Distributive Property A(B+C)=AB=AC Rigtht Distibutive property (A+B)C=AC+BC Asssosiative Property of Scalar Multiplication c(AB)=(cA)B=a(cB) Determinant of a Matrix
The determinant of a matrix is the difference of the products of the entries on the diagonals.
Notes 4.4: Inverses and Identities.
The number one is the multiplicative identity for all real numbers because 1*b=b and b*1=b. The "n X n" Identity Matrix is that matrix that has 1's o the main diagonal and 0's everywhere else. ex. for a 2 X 2 matrix :
||
1
0
0
1
If B is any matrix and I is the identity matrix, then IB=B and BI=B
two n x n matrices are inverses of each other if their product (in both orders) is the n x n identity matrix.
Ex.
[3 -1][2 1] = [1 0]
[-5 2] [5 3]= [0 1]
Notes 4.5: Solving Equations with Matrices.
1. Find the determinent of [A]
2. Muptiply [A] the reciprocal of the det.
3. In the new Matrix [A^(-1)], switch the a&d positions and the c&b signs.
4. Multiply [A^(-1)] by [B] to get [X]
Data of sales records, such as for CD’s or DVD’s,
can be organized into a matrix to represent the data.
Real Life Application 4.2
Matrices can be set up to determine the cost of
operation or the cost of equipment for a business.
Because multiplying matrices requires only the
multiplication of corresponding sectors of the matrix,
cost and quantity can be multiplied together for cost.
Real Life Application 4.3
The area of a triangle can be determined with
matrices, which can be used in architecture or landscaping.
The matrix is set up where the first column of entries is the x-value
of the triangle's coordinates, while the second column is the y-value.
The third column is then filled with 1's. By finding the determinant, the
area can be found after dividing by 2 and using the absolute value.
Example Problems 4.1
4
9
17
12
+
16
11
3
8
=
20
20
20
20
Example Problems 4.2
Vocab 4.1
Matrix- a rectangular arrangement of numbers in rows and columns. [ 3 4 ] Dimensions- the number m of rows of a matrix by the number n of columns of the matrix, written m x n. [ 3 4 ]= 1 x 2 Entries- the numbers in a matrix. 3, 4 Equal- when the dimensions are the same and the entries in corresponding positions are equal in the matrices. Scalar- a real number by which you multiply a matrix. 2[ 3 4 ] Scalar Multiplication- the process of multiplying each entry in a matrix by a scalar.
Vocab 4.3
Determinant- a real number associated with any square matrix A, denoted by det A or by the absolute value of A. Cramer's Rule- a method for solving a system of linear equations which uses determinants of matrices. Coefficient Matrix- a matrix consisting of the coefficients of the variables in a set of linear equations.
Vocab 4.4
Identity Matrix- the matrix that has 1's on the main diagonal and 0's elsewhere. Inverses- two n x n matrices are inverses of each other if their product (in both orders) is the n x n identity matrix.
Vocab 4.5 Matrix of Variables- the matrix of variables of the linear system ax + by = e, cx + dy = f is [x over y]
Italic
. Matrix of Constants- the matrix of constants of the linear system ax + by = e, cx + dy = f is [e over f].
Notes 4-5: Katie Conner
Example Problems: Cody Windsor
Vocab: Austin Gibbons
editor/ video: Cheyenne Bohlen
notes 1-3:Haley Brandt
A matrix is a rectangular arrangement of numbers in rows ad columns. For instance, matrix A below has two rows and three columns. The dimensions of this matrix are 2x3 (read 2 by 3). The numbers in a matrix are its entries. In matrix A, the entry in the second row, third column is 5.
Properties of matrix operations
Let A, B, and C be matrices of the same dimensions and let d be scalar.
Associative Property of Addition (A+B)+C=A+(B+C)
Commutative Property of Addition A+B=B+A
Multiplication of a sum or difference of a matrix follows the distributive property.
Distributive Property of Addition d(A+B)=dA=dB
Distributive Property of Subtraction d(A-B)=dA-dB
Multiplying Matrices
The product of two matrices is only defined if the number of columns in A is equal to the number of rows in B. If A has a dimension of mxn and B had a dimension of nxp then the product of AB is mxp matrix.
There is a specific pattern for multiplying matrices. The first number of the first row of the A matrix is multiplied by the first number of the first column of the B matrix.
Next multiply the second number of the first row of the A matrix by the second number of the first column of the B matrix.
Then add the products ae and bg. This is the first entry in your new matrix, C.
ae+bg=i
Next multiply the first number of the first row in matrix A by the first number in the second column of matrix B.
Then multiply the second number of the first row in matrix A by the second number in the second row of matrix B.
Add the products af and bh. This becomes the second number of the first row in your new C matrix.
af+bh=j
Take the first number of the second row of matrix A and multiply it by the first number in the first column of matrix B
Next take the second number of the second row of matrix A and multiply it by the second number of the first column of the B matrix.
Next take the product ce and add it to dg. This is the third number in your new matrix.
Take the first number of the second row of matrix A and multiply it by the first number of the second column of matrix B.
Take the second number of the second row of matrix A and multiply it by the second number of the second column of matrix B.
Now add the products cf and dh. This is the last number of your matrix.
Multiplication Properties of matrices
Assosiative Property of matrix multiplication A(BC)=(AB)C
Left Distributive Property A(B+C)=AB=AC
Rigtht Distibutive property (A+B)C=AC+BC
Asssosiative Property of Scalar Multiplication c(AB)=(cA)B=a(cB)
Determinant of a Matrix
The determinant of a matrix is the difference of the products of the entries on the diagonals.
Notes 4.4: Inverses and Identities.
The number one is the multiplicative identity for all real numbers because 1*b=b and b*1=b. The "n X n" Identity Matrix is that matrix that has 1's o the main diagonal and 0's everywhere else. ex. for a 2 X 2 matrix :
||
two n x n matrices are inverses of each other if their product (in both orders) is the n x n identity matrix.
Ex.
[3 -1][2 1] = [1 0]
[-5 2] [5 3]= [0 1]
Notes 4.5: Solving Equations with Matrices.
1. Find the determinent of [A]
2. Muptiply [A] the reciprocal of the det.
3. In the new Matrix [A^(-1)], switch the a&d positions and the c&b signs.
4. Multiply [A^(-1)] by [B] to get [X]
[A]=
[ 2 -1]
[2 -1]
[B]=
[1 2]
[4 4]
Det of [A] =-4
[A^(-1)]=(-1/4)[2 -1]=[-0.5 0.25]= [0.25 -0.25]
....................[2 -1] =[-0.5 0.25]=[0.5 -0.5]
[A^(-1)][B]=[X]
[X]=[-0.75 -0.5]
......[-1.5 -1]
Real Life Application 4.1
Data of sales records, such as for CD’s or DVD’s,
can be organized into a matrix to represent the data.
Real Life Application 4.2
Matrices can be set up to determine the cost of
operation or the cost of equipment for a business.
Because multiplying matrices requires only the
multiplication of corresponding sectors of the matrix,
cost and quantity can be multiplied together for cost.
Real Life Application 4.3
The area of a triangle can be determined with
matrices, which can be used in architecture or landscaping.
The matrix is set up where the first column of entries is the x-value
of the triangle's coordinates, while the second column is the y-value.
The third column is then filled with 1's. By finding the determinant, the
area can be found after dividing by 2 and using the absolute value.
Example Problems 4.1
Vocab 4.1
Matrix- a rectangular arrangement of numbers in rows and columns. [ 3 4 ]
Dimensions- the number m of rows of a matrix by the number n of columns of the matrix, written m x n. [ 3 4 ]= 1 x 2
Entries- the numbers in a matrix. 3, 4
Equal- when the dimensions are the same and the entries in corresponding positions are equal in the matrices.
Scalar- a real number by which you multiply a matrix. 2[ 3 4 ]
Scalar Multiplication- the process of multiplying each entry in a matrix by a scalar.
Vocab 4.3
Determinant- a real number associated with any square matrix A, denoted by det A or by the absolute value of A.
Cramer's Rule- a method for solving a system of linear equations which uses determinants of matrices.
Coefficient Matrix- a matrix consisting of the coefficients of the variables in a set of linear equations.
Vocab 4.4
Identity Matrix- the matrix that has 1's on the main diagonal and 0's elsewhere.
Inverses- two n x n matrices are inverses of each other if their product (in both orders) is the n x n identity matrix.
Vocab 4.5
Matrix of Variables- the matrix of variables of the linear system ax + by = e, cx + dy = f is [x over y]
Matrix of Constants- the matrix of constants of the linear system ax + by = e, cx + dy = f is [e over f].